Problem: Select all polynomials that have $(x-3)$ as a factor. Choose all answers that apply: Choose all answers that apply: (Choice A) A $A(x)=x^3-2x^2-4x+3$ (Choice B) B $B(x)=x^3+3x^2-2x-6$ (Choice C) C $C(x)=x^4-2x^3-27$ (Choice D) D $D(x)=x^4-20x-21$
Solution: The following statements are equivalent: $(x-3)$ is a factor of $p(x)$ $p(x)$ is divisible by $(x-3)$ The remainder of $\dfrac{p(x)}{x-3}$ is $0$ We can use the polynomial remainder theorem to solve this problem: For a polynomial $p(x)$ and a number $a$, the remainder on division by $x-a$ is $p(a)$. According to the theorem, the remainder when $p(x)$ is divided by $(x-{3})$ is equal to $p({3})$. So to check each polynomial if it has $(x-3)$ as a factor, we need to check if that polynomial's value at ${x=3}$ is zero. $\begin{aligned} A({3})&=0 \\\\ B({3})&=42 \\\\ C({3})&=0 \\\\ D({3})&=0 \end{aligned}$ In conclusion, the following polynomials have $(x-3)$ as a factor: $A(x)=x^3-2x^2-4x+3$ $C(x)=x^4-2x^3-27$ $D(x)=x^4-20x-21$